radially equidistant "melting" of image onto spheroid (head) - reverse of equidistant azimuthal projection
A solution based on Python (image and svg found here) :



The script :

import bpy
import bmesh
from math import *
from mathutils import Vector

#Reposition the mesh vertices calculating a 'reverse of equidistant azimuthal projection'
def reverseOfEquidistantAzimuthalProjection( model, propFactor = 1 ):

    rad90 = radians(90.0) #A way to have pi / 2

    modelData = model.data

    #Calculates vertices boundings (relative to vertices coordinates, because of the case of a not centered mesh)
    minX = min( [v.co[0] for v in modelData.vertices] )
    maxX = max( [v.co[0] for v in modelData.vertices] )
    minY = min( [v.co[1] for v in modelData.vertices] )
    maxY = max( [v.co[1] for v in modelData.vertices] )

    #Center in X and Y, we consider here a flat mesh in Z
    centerX = (minX + maxX) / 2.0
    centerY = (minY + maxY) / 2.0
    center = Vector( [centerX, centerY, 0] )

    #Radius of the projection
    baseRadius = (maxX - centerX)

    for v in modelData.vertices:
        pos = Vector( v.co )

        #Position relative to center        
        posXY = pos.xy - center.xy

        #Proportionality of the vertex relative to the radius
    #Between 0 and 100% where 0 corresponds to north pole and 100 to south
        prop = posXY.length / baseRadius

        #Conversion to -90 (south) / +90 (north) (in rads)
        angle = rad90 - (prop * pi) 

        #Relocation in X and Y : cos(angle) gives the radius of the cut
        posXY.normalize()
        posXY = posXY * baseRadius * cos(angle) * propFactor

        #Set the vertices new coordinates
        v.co[0] = center.x + posXY.x
        v.co[1] = center.y + posXY.y
        #In Z : keeps the eventual Z value and place the point relatively to that
        v.co[2] = v.co[2] + sin( angle ) * baseRadius * propFactor


model = bpy.context.object

reverseOfEquidistantAzimuthalProjection( model )


The first experiment is based on a circle extruded and cut :


I have also try to use the svg symbol of the United Nations. Some difficulties :


The curves are in a lot of parts (you may have to select them all, convert to mesh and join them)
Edges are made 'as they come', so applying the script gives some unwanted results (some edges may have to be removed manually)